How to Add Fractions with Different Denominators

Why you cannot just add across

The first instinct is usually to add the tops and add the bottoms: 1/2 + 1/3 = 2/5. It is worth seeing clearly why that does not work, because understanding the reason makes the real method feel less arbitrary.

1/2 is a half. 1/3 is a bit less than a half. Together they must be nearly one whole. But 2/5 is less than a half — smaller than what you started with. Adding two positive amounts cannot make the total smaller, so the method must be wrong.

The problem is that halves and thirds are different-sized pieces. You cannot add them directly for the same reason you cannot add 3 metres and 4 feet without converting first.

Step 1 — find a common denominator

You need both fractions cut into the same size pieces. The denominators are 2 and 3, and the smallest number both divide into is 6.

A quick way to find one: multiply the two denominators together. 2 × 3 = 6. That always gives a common denominator, though not always the smallest one.

For 4 and 6, multiplying gives 24, which works — but 12 also works and keeps the numbers smaller. Either is correct; the smaller one just means less simplifying later.

Step 2 — rewrite both fractions

To turn 1/2 into sixths, ask what you multiplied 2 by to get 6. The answer is 3, so multiply the top by 3 as well: 1/2 becomes 3/6.

For 1/3: you multiplied 3 by 2 to get 6, so multiply the top by 2 as well. 1/3 becomes 2/6.

Multiplying top and bottom by the same number never changes a fraction's value — you are cutting the same amount into more, smaller pieces. Forgetting to multiply the top is the most common error at this step, and it silently changes the value.

Step 3 — add the numerators

Now that the pieces are the same size, adding is straightforward. 3/6 + 2/6 = 5/6.

The denominator does not change. You are counting sixths, and adding three of them to two of them gives five of them. The size of each piece has not altered.

Adding the denominators as well is the other frequent slip here, and it gives 5/12, which is half the right answer.

Step 4 — simplify if you can

5/6 cannot be reduced, since 5 and 6 share no common factor. If your answer had come out as 4/6, you would divide both by 2 to get 2/3.

Most courses expect the answer in lowest terms, so it is worth checking even when the answer looks tidy.

Adding mixed numbers

For something like 2¼ + 1⅓, you have two options.

The reliable one is to convert both to improper fractions first — 9/4 and 4/3 — then add as usual and convert back at the end.

The quicker one is to add the whole numbers and the fractions separately: 2 + 1 = 3, and ¼ + ⅓ = 7/12, giving 3 7/12.

The second is faster but has a trap. If the fraction part comes out greater than 1, you have to carry it into the whole number. ¾ + ⅔ = 17/12, which is 1 5/12, so that 1 gets added to your whole-number total.